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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 1680a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1680.c3 | 1680a1 | \([0, -1, 0, -36, 96]\) | \(20720464/105\) | \(26880\) | \([2]\) | \(128\) | \(-0.30352\) | \(\Gamma_0(N)\)-optimal |
| 1680.c2 | 1680a2 | \([0, -1, 0, -56, 0]\) | \(19307236/11025\) | \(11289600\) | \([2, 2]\) | \(256\) | \(0.043052\) | |
| 1680.c1 | 1680a3 | \([0, -1, 0, -656, -6240]\) | \(15267472418/36015\) | \(73758720\) | \([2]\) | \(512\) | \(0.38963\) | |
| 1680.c4 | 1680a4 | \([0, -1, 0, 224, -224]\) | \(604223422/354375\) | \(-725760000\) | \([2]\) | \(512\) | \(0.38963\) |
Rank
sage: E.rank()
The elliptic curves in class 1680a have rank \(1\).
Complex multiplication
The elliptic curves in class 1680a do not have complex multiplication.Modular form 1680.2.a.a
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
